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DCDS

In the present paper we study the local behavior of non-negative weak solutions of a wide class of doubly non linear degenerate parabolic equations. We show, in particular, some lower pointwise estimates of such solutions in terms of suitable sub-potentials (dictated by the structure of the equation) and an alternative form of the Harnack inequality.

CPAA

We consider non-homogeneous, singular ($ 0 < m < 1 $) porous medium type equations with a non-negative Radon-measure $\mu$ having finite total mass $\mu(E_T)$ on the
right-hand side. We deal with a Cauchy-Dirichlet problem for these type of equations, with homogeneous boundary conditions on the parabolic boundary of the domain $E_T$, and we establish the existence of a solution in the sense of distributions.
Finally, we show that
the constructed solution satisfies linear pointwise estimates via linear
Riesz potentials.

keywords:
existence
,
boundedness.
,
Singular porous medium equations
,
very weak solutions
,
Riesz potential

DCDS-S

Let $u$ be a non-negative super-solution to a $1$-dimensional
singular parabolic equation of $p$-Laplacian type
($1< p <2$). If $u$ is
bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$,
then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in $\mathbb R\times[T/2,T]$. This fact, stated
quantitatively in Proposition 1.2,
is a ``sidewise spreading of positivity'' of solutions
to such singular equations, and can be considered as a
form of Harnack inequality. The proof of such an effect is based on geometrical ideas.

DCDS-B

The local positivity of solutions to logarithmically
singular diffusion equations is investigated in some
open space-time domain $E\times(0,T]$. It is
shown that if at some time level $t_o\in(0,T]$ and some point
$x_o\in E$ the solution $u(\cdot,t_o)$ is not
identically zero in a neighborhood of $x_o$, in
a measure-theoretical sense, then it is strictly
positive in a neighborhood of $(x_o, t_o)$. The precise
form of this statement is by an intrinsic Harnack-type
inequality, which also determines the size of such
a neighborhood.

## Year of publication

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